Question

The maximum value of $|z|$ when the complex number $z$ satisfies the condition $\left|z+\frac{2}{z}\right|$ = 2 is

• A. $\sqrt{3}$
• B. $\sqrt{3}+\sqrt{2}$
• C. $\sqrt{3}+1$
• D. $\sqrt{3}-1$

Answer 1

Answer: (C)

We have the identity,
$|z|=\left |z+\frac{2}{z}-\frac{2}{z}\right| \leq\left |z+\frac{2}{z}\right|+\left|\frac{2}{z}\right|$
$\Rightarrow |z| \leq 2+\frac{2}{|z|}$
$\Rightarrow |z|^{2} \leq 2|z|+2$
$\Rightarrow |z|^{2}-2|z|+1 \leq 3$
$\Rightarrow (|z|-1)^{2} \leq 3$
$\Rightarrow -\sqrt{3} \leq |z|-1 \leq \sqrt{3}$
$\Rightarrow -\sqrt{3}+1 \leq |z| \leq \sqrt{3}+1$
Hence, the maximum value $=\sqrt{3}+1$